metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊1C4, Q16⋊2F5, D10.3D8, Dic5.17SD16, C8.3(C2×F5), C40.3(C2×C4), C40⋊C4⋊2C2, (C5×Q16)⋊1C4, C5⋊2(D8⋊2C4), (C4×D5).24D4, C5⋊2C8.15D4, C8.F5⋊3C2, C4.5(C22⋊F5), Q8.D10.2C2, C20.5(C22⋊C4), (C8×D5).11C22, C10.9(D4⋊C4), C2.10(D20⋊C4), SmallGroup(320,245)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊1C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a37b >
Subgroups: 346 in 58 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×3], C22 [×2], C5, C8, C8, C2×C4 [×3], D4 [×2], Q8, D5 [×2], C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, F5, D10, D10, C4.Q8, M5(2), C4○D8, C5⋊2C8, C40, C4×D5, C4×D5, D20 [×2], C5×Q8, C2×F5, D8⋊2C4, C5⋊C16, C8×D5, D40, Q8⋊D5, C5×Q16, C4⋊F5, Q8⋊2D5, C8.F5, C40⋊C4, Q8.D10, D40⋊1C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, F5, D4⋊C4, C2×F5, D8⋊2C4, C22⋊F5, D20⋊C4, D40⋊1C4
Character table of D40⋊1C4
class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5 | 8A | 8B | 8C | 10 | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 40A | 40B | |
size | 1 | 1 | 10 | 40 | 2 | 8 | 10 | 40 | 40 | 4 | 4 | 10 | 10 | 4 | 20 | 20 | 20 | 20 | 8 | 16 | 16 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | i | -i | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | i | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ9 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
ρ12 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
ρ13 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -√-2 | -√-2 | √-2 | √-2 | -2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
ρ14 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | √-2 | √-2 | -√-2 | -√-2 | -2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
ρ15 | 4 | 4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
ρ16 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
ρ17 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
ρ18 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -2√-2 | 2√-2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊2C4 |
ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 2√-2 | -2√-2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊2C4 |
ρ21 | 8 | 8 | 0 | 0 | -8 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D20⋊C4, Schur index 2 |
ρ22 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√10 | √10 | orthogonal faithful, Schur index 2 |
ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √10 | -√10 | orthogonal faithful, Schur index 2 |
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)
(2 28 10 4)(3 15 19 7)(5 29 37 13)(6 16)(8 30 24 22)(9 17 33 25)(11 31)(12 18 20 34)(14 32 38 40)(23 35 39 27)(26 36)(41 74 45 62)(42 61 54 65)(43 48 63 68)(44 75 72 71)(46 49 50 77)(47 76 59 80)(51 64 55 52)(53 78 73 58)(56 79 60 67)(57 66 69 70)
G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,74,45,62)(42,61,54,65)(43,48,63,68)(44,75,72,71)(46,49,50,77)(47,76,59,80)(51,64,55,52)(53,78,73,58)(56,79,60,67)(57,66,69,70)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,74,45,62)(42,61,54,65)(43,48,63,68)(44,75,72,71)(46,49,50,77)(47,76,59,80)(51,64,55,52)(53,78,73,58)(56,79,60,67)(57,66,69,70) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53)], [(2,28,10,4),(3,15,19,7),(5,29,37,13),(6,16),(8,30,24,22),(9,17,33,25),(11,31),(12,18,20,34),(14,32,38,40),(23,35,39,27),(26,36),(41,74,45,62),(42,61,54,65),(43,48,63,68),(44,75,72,71),(46,49,50,77),(47,76,59,80),(51,64,55,52),(53,78,73,58),(56,79,60,67),(57,66,69,70)])
Matrix representation of D40⋊1C4 ►in GL8(𝔽241)
0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
240 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 240 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 222 | 19 | 0 | 0 |
0 | 0 | 0 | 0 | 222 | 222 | 0 | 0 |
0 | 0 | 0 | 0 | 192 | 114 | 38 | 95 |
0 | 0 | 0 | 0 | 13 | 129 | 137 | 0 |
234 | 124 | 117 | 7 | 0 | 0 | 0 | 0 |
117 | 0 | 124 | 7 | 0 | 0 | 0 | 0 |
234 | 7 | 124 | 0 | 0 | 0 | 0 | 0 |
0 | 7 | 117 | 124 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 192 | 114 | 38 | 95 |
0 | 0 | 0 | 0 | 192 | 114 | 0 | 95 |
0 | 0 | 0 | 0 | 222 | 19 | 0 | 0 |
0 | 0 | 0 | 0 | 135 | 59 | 116 | 176 |
0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
0 | 0 | 0 | 0 | 49 | 127 | 0 | 146 |
0 | 0 | 0 | 0 | 206 | 194 | 137 | 0 |
G:=sub<GL(8,GF(241))| [0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,222,222,192,13,0,0,0,0,19,222,114,129,0,0,0,0,0,0,38,137,0,0,0,0,0,0,95,0],[234,117,234,0,0,0,0,0,124,0,7,7,0,0,0,0,117,124,124,117,0,0,0,0,7,7,0,124,0,0,0,0,0,0,0,0,192,192,222,135,0,0,0,0,114,114,19,59,0,0,0,0,38,0,0,116,0,0,0,0,95,95,0,176],[0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,49,206,0,0,0,0,0,240,127,194,0,0,0,0,0,0,0,137,0,0,0,0,0,0,146,0] >;
D40⋊1C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_1C_4
% in TeX
G:=Group("D40:1C4");
// GroupNames label
G:=SmallGroup(320,245);
// by ID
G=gap.SmallGroup(320,245);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,675,794,80,1684,851,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^37*b>;
// generators/relations
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